What is a variety? It is the set of common zeroes for a set of polynomials. For example, for the set of polynomials ![\{x+y,x-y\}\in\Bbb{R}[x,y]](https://s0.wp.com/latex.php?latex=%5C%7Bx%2By%2Cx-y%5C%7D%5Cin%5CBbb%7BR%7D%5Bx%2Cy%5D&bg=FFFFFF&fg=111111&s=0&c=20201002)
Now what is a projective variety? Simply put, it is the common set of zeroes of polynomials in which a one-dimensional subspace is effectively considered one point. Hence, for the variety to be well-defined, if one point of a one-dimensional subspace satisies the variety, every point of the one-dimensional subspace has to satisfy that variety. Confused?
Take the polynomial ![x+y+z\in\Bbb{C}[x,y,z]](https://s0.wp.com/latex.php?latex=x%2By%2Bz%5Cin%5CBbb%7BC%7D%5Bx%2Cy%2Cz%5D&bg=FFFFFF&fg=111111&s=0&c=20201002)
![x+y+z-1\in\Bbb{C}[x,y,z]](https://s0.wp.com/latex.php?latex=x%2By%2Bz-1%5Cin%5CBbb%7BC%7D%5Bx%2Cy%2Cz%5D&bg=FFFFFF&fg=111111&s=0&c=20201002)
But why? Why would you want to consider a whole line as one point? When you watch the world from your little nest, every line running along your ine of sight becomes a point. Hence, athough it may be a line in “reality” (whatever this means), for you it is a point. This is the origin of projective geometry, although things have gotten sightly complicated since then.
